1 / 18

Weighted counting of k- matchings is #W[1]- hard

Weighted counting of k- matchings is #W[1]- hard. Markus Bläser, Radu Curticapean Saarland University, Computational Complexity Group. c ounting ((perfect) matchings). since 1976 and counting. for biadjacency matrix of bipartite . considered intractable. poly-time computable.

sumana
Download Presentation

Weighted counting of k- matchings is #W[1]- hard

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. Weightedcountingofk-matchingsis #W[1]-hard Markus Bläser, Radu Curticapean Saarland University, Computational Complexity Group

  2. counting ((perfect) matchings) since 1976 and counting. for biadjacency matrix of bipartite considered intractable poly-time computable 1967#Planar-PerfMatch (Fisher, Kasteleyn) 1976definition of , hardness of (Valiant) 1989FPRAS for (Jerrum, Sinclair) 2004 parameterized counting complexity (Flum, Grohe) 2007 on planar G of is #P-hard. (Xia et al.)

  3. parameterized counting • parameterized counting problems • on input , decide count something • typically solvable in time or time • count -vertex covers • count -cliques: ) • count -matchings: ) • #W[1]can be defined as closure of under • fpt-turing reduction • solves in fpt-time with oracle for • queries have

  4. known results „simple“ substructure in:graph, par: out: for any MSOL formula in:graph par: treewidth out: sets with in:graph par: genus out: „simple“ graph

  5. our result status of? • „hardness of permanent“ in fpt-world • decision version is -hard. in:edge-weighted bipartite G, out: proof by series of reductions

  6. partial path-cycle covers • k-partial cyclecover • of cycles • vertex-disjoint • edges in total • k-partial path-cycle cover • of pathsand cycles • … • …

  7. reduction chain in:weightedbipartite G, out: in:weighteddigraph G, out: in:digraph G, out:

  8. matchings path-cycle covers standard reduction G S(G) split in out implies

  9. reduction chain in:weightedbipartite G, out: in:weighteddigraph G, out: in:digraph G, out:

  10. path-cyclecovers cycle covers b b b b b b • recipe for path-cycle covers in • path-cycle cover in as core • at path ends:add nothingor • at isolated vts:addnothing or or -b -b -b -b -b -b -b b b • interpolation along allows to track out paths analysis via path-cycle polynomial transform by gadgets

  11. reduction chain in:digraph G, out: in:digraph G, type , out: in:graph G, out:

  12. cycle-like structures cycle norepeatingvts. CW closedwalk UCW for tuple , letcyclic shiftsof

  13. types of UCWs #visits of at 2 3 1 type 1 size (omitting 0s is fine) 0 1 size size similar to types of cyclic walks defined by Flum, Grohe

  14. reduction from cliques • In G, replace all edges by , and add . • Want to count induced subgraphs isomorphic to . • Consider set of UCWs of type in G. k • Partition according to visited vertices • is some graph on vertices. • If , then „same“ UCWs in and . • Partition according to (isomorphism type of) proof adapted from Flum, Grohe

  15. within partition class of H • UCWs of type in G • induced in H k vertices • UCWs of type in

  16. typed UCWs cliques UCWs of type in oracle call UCWs of type ioracle call isomorphic copies of in wanted prove that this column is lin. indep. for some proof adapted from Flum, Grohe

  17. reductionchain in:weightedbipartite G, out: in:graph G, out: in:weighteddigraph G, out: in:digraph G, type , out: in:digraph G, out:

  18. futurework in:weightedbipartite G, out: in:graph G, out: in:weighteddigraph G, out: in:digraph G, type , out: in:digraph G, out:

More Related